Exploring Roots of Quadratics
The general quadratic has equation . All of the possible roots (real and imaginary) of this polynomial can be found using the quadratic formula , in which the radicand can tell us which roots are real and which are imaginary. LetÕs explore how the variable ÒbÓ changes the polynomial and the resulting roots. Below is a graph of a quadratic where ÒbÓ is the altering variable.
As you can see, the vertex is traveling along a path that looks much like another parabola itself. The vertex travels through the bottom two quadrants depending on the sign of ÒbÓ. If ÒbÓ is positive, the vertex travels to the right and in this case, parabola will be on the positive side of the x-axis. If ÒbÓ is negative, the vertex travels to the left and will be on the negative side of the x-axis.
Now letÕs consider graphing the parabola in the x-b plane. This means that our ÒbÓ value will become our dependent variable while our ÒxÓ value remains the independent variable.
By solving the general parabola for ÒbÓ, we will obtain a rational equation.
When c=1, our graph of this equation looks like this:
From the above graph, we can conclude that when c=1, the range of the function includes all real numbers except when . We can also see that whenever our x-value is negative, our y-value (our actual ÒbÓ) is also negative and vice versa when our x-value is positive.
If we take any particular value of b, say b = 3, and overlay this equation on the graph we add a line parallel to the x-axis. If it intersects the curve in the x-b plane the intersection points correspond to the roots of the original equation for that value of b. We have the following graph.
The graph above shows that when b=3 in our original quadratic equation , the roots for this equation are approximately x=-.4 and x=-2.6. From the above, we have now graphed our equation in Òx-b planeÓ. This means that instead of looking at x=intercepts for our roots, we are now looking at the intersections of our rational polynomial with the horizontal line b=3.
Now letÕs look at this graph in motion.
For each value of b we select, we get a horizontal line. It is clear on a single graph that we get two negative real roots of the original equation when b > 2, one negative real root when b = 2, no real roots for -2 < b < 2, One positive real root when b = -2, and two positive real roots when b < -2.
Consider the case when c = - 1 rather than + 1.
As you can see, we now have one positive root at approximately x=.1 and another at approximately x=-7.9. LetÕs try other values of ÒcÓ.
How about c=0?
When c=0, we only have one real negative root. Since we still have a quadratic, we must have two roots. In this case, one is real and one is imaginary.